Kronecker-Weber plus epsilon

Abstract

Abstract. We say that a group is almost abelian if every commutator is central and squares to the identity. Now let G be the Galois group of the algebraic closure of the field Q of rational numbers in the field of complex numbers. Let G ab +ǫ be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups and let Q ab +ǫ /Q be the corresponding Galois extension. We prove that Q ab +ǫ is generated by the roots of unity, the fourth roots of the rational primes and the square roots of certain algebraic sine-monomials. The inspiration for the paper came from recent studies of algebraic Γ-monomials by P. Das and by S. Seo.

All double coverings of cyclotomic fields

“…Different from the cases in [2] and [5], but similar to the case in [6], the homomorphism now is not injective. Its kernel corresponds to the field K 1+ ab , which we have determined in the proposition.…”

“…Its kernel corresponds to the field K 1+ ab , which we have determined in the proposition. By comparing the Z/2Z-dimensions of the two sides and using Anderson's main result in [2] we show that the induced homomorphism…”

“…Let Q ab be the maximal abelian extension of the rational number field Q in the complex number field C. Let Q ab+ be the composite of all double coverings of the extension Q ab /Q. G. Anderson [2] exhibit an explicit description of Q ab+ as explicit as that provided for Q ab by the Kronecker-Weber theorem. Bae and Yin [5] explicitly describe all (q −1)-th coverings of the maximal abelian extension of F q (T ).…”

Anderson’s double complex and gamma monomials for rational function fields

“…But the calculation of D( q−1 e(a)) would be too complicated to take. About this question, we refer the reader to Remark 4.4.2 in [An2].…”

Sign functions of imaginary quadratic fields and applications

“…These cohomology groups naturally appear in many settings. See for instance Anderson's theory of epsilon extensions and it's analog for function fields in [2] and [3]. Let us remark that U s (m) is naturally a Gal(k m,s /k)-module.…”